Computer system and method using a kautz-like digraph to interconnect computer nodes and having control back channel between nodes

ABSTRACT

Computer system and method using a Kautz-like digraph to interconnect computer nodes and having control back channel between nodes. A multinode computing system includes a large plurality of computing nodes interconnected via a Kautz topology having order O, diameter n, and degree k. The order equals (k+1)k n−1 ; The data interconnections from a node x to a node y in the topology satisfy the relationship y=(−x*k−j) mod O, where 1≦j≦k; and each x,y pair includes a unidirectional control link from node y to node x to convey flow control and error information from a receiving node y to a transmitting node x.

CROSS-REFERENCE TO RELATED APPLICATIONS

This application is related to the following U.S. patent applications, the contents of which are incorporated herein in their entirety by reference:

U.S. patent application Ser. No. 11/335,421, filed Jan. 19, 2006, entitled SYSTEM AND METHOD OF MULTI-CORE CACHE COHERENCY;

U.S. patent application Ser. No. 11/594,416, filed Nov. 8, 2006, entitled COMPUTER SYSTEM AND METHOD USING EFFICIENT MODULE AND BACKPLANE TILING TO INTERCONNECT COMPUTER NODES VIA A KAUTZ-LIKE DIGRAPH;

U.S. patent application Ser. No. 11/594,426, filed Nov. 8, 2006, entitled SYSTEM AND METHOD FOR PREVENTING DEADLOCK IN RICHLY-CONNECTED MULTI-PROCESSOR COMPUTER SYSTEM USING DYNAMIC ASSIGNMENT OF VIRTUAL CHANNELS;

U.S. patent application Ser. No. 11/594,421, filed Nov. 8, 2006, entitled LARGE SCALE MULTI-PROCESSOR SYSTEM WITH A LINK-LEVEL INTERCONNECT PROVIDING IN-ORDER PACKET DELIVERY;

U.S. patent application Ser. No. 11/594,442, filed Nov. 8, 2006, entitled MESOCHRONOUS CLOCK SYSTEM AND METHOD TO MINIMIZE LATENCY AND BUFFER REQUIREMENTS FOR DATA TRANSFER IN A LARGE MULTI-PROCESSOR COMPUTING SYSTEM;

U.S. patent application Ser. No. 11/594,427, filed Nov. 8, 2006, entitled REMOTE DMA SYSTEMS AND METHODS FOR SUPPORTING SYNCHRONIZATION OF DISTRIBUTED PROCESSES IN A MULTIPROCESSOR SYSTEM USING COLLECTIVE OPERATIONS;

U.S. patent application Ser. No. 11/594,420, filed Nov. 8, 2006, entitled SYSTEM AND METHOD FOR ARBITRATION FOR VIRTUAL CHANNELS TO PREVENT LIVELOCK IN A RICHLY-CONNECTED MULTI-PROCESSOR COMPUTER SYSTEM;

U.S. patent application Ser. No. 11/594,441, filed Nov. 8, 2006, entitled LARGE SCALE COMPUTING SYSTEM WITH MULTI-LANE MESOCHRONOUS DATA TRANSFERS AMONG COMPUTER NODES;

U.S. patent application Ser. No. 11/594,405, filed Nov. 8, 2006, entitled SYSTEM AND METHOD FOR COMMUNICATING ON A RICHLY CONNECTED MULTI-PROCESSOR COMPUTER SYSTEM USING A POOL OF BUFFERS FOR DYNAMIC ASSOCIATION WITH A VIRTUAL CHANNEL;

U.S. patent application Ser. No. 11,594,443, filed Nov. 8, 2006, entitled RDMA SYSTEMS AND METHODS FOR SENDING COMMANDS FROM A SOURCE NODE TO A TARGET NODE FOR LOCAL EXECUTION OF COMMANDS AT THE TARGET NODE;

U.S. patent application Ser. No. 11/594,447, filed Nov. 8, 2006, entitled SYSTEMS AND METHODS FOR REMOTE DIRECT MEMORY ACCESS TO PROCESSOR CACHES FOR RDMA READS AND WRITES; and

U.S. patent application Ser. No. 11/594,446, filed Nov. 8, 2006, entitled SYSTEM AND METHOD FOR REMOTE DIRECT MEMORY ACCESS WITHOUT PAGE LOCKING BY THE OPERATING SYSTEM.

BACKGROUND OF THE INVENTION

1. Field of the Invention

The present invention relates to massively parallel computing systems and, more specifically, to computing systems in which computing nodes are interconnected via a Kautz-like topology for data links and which include a control back channel in the opposite direction.

2. Discussion of Related Art

Massively parallel computing systems have been proposed for scientific computing and other compute-intensive applications. The computing system typically includes many nodes, and each node may contain several processors. Various forms of interconnect topologies have been proposed to connect the nodes, including Hypercube topologies, butterfly and omega networks, tori of various dimensions, fat trees, and random networks.

One problem that has been observed with certain architectures is the issue of scalability. That is, due to inherent limitations, certain architectures are not easily scalable in any practical way. For example, one cannot simply add processing power by including another module of computing nodes into the system, or more commonly, the expense and/or performance of the network becomes unacceptable as it grows larger. Moreover, different sized systems might need totally different module designs. For example, hypercube topologies had nodes in which the number of ports or links was dependent on the overall size of the system. Thus a node made for one size system could not be used, as a general matter, on a system with a different size.

Another problem that has been observed is that of routing the connections among nodes. Large systems typically cannot be fully connected because of inherent difficulty in routing. Thus switching architectures have been proposed, but these introduce latency from the various “hops” among nodes that may be necessary for two arbitrary nodes to communicate with one another. Reducing this latency is desirable but has proven difficult.

SUMMARY

The invention provides computer system and method using a Kautz-like digraph to interconnect computer nodes and having control back channel between nodes.

Under one aspect of the invention, a multinode computing system includes a large plurality of computing nodes interconnected via a Kautz topology having order O, diameter n, and degree k. The order equals (k+1)k^(n−1); The data interconnections from a node x to a node y in the topology satisfy the relationship y=(−x*k−j) mod O, where 1≦j≦k; and each x,y pair includes a unidirectional control link from node y to node x to convey flow control and error information from a receiving node y to a transmitting node x.

Under another aspect of the invention, a receiving node y transmits control packets on a control link to transmitting node x to identify the last correctly received data packet, and to identify whether an error has been detected in transmission.

Under another aspect of the invention, a transmitting node x stores transmitted packets and keeps them available for replay in response to control messages on the control link.

Under another aspect of the invention, a receiving node y transmits buffer status information to a transmitting node x to identify buffer availability of downstream computing nodes.

Under another aspect of the invention, a transmitting node x includes logic to transmit a packet downstream only if all necessary buffers are available.

BRIEF DESCRIPTION OF THE DRAWINGS

In the Drawing,

FIGS. 1A-C depict Kautz topologies of various order, degree and diameter;

FIG. 1D depicts a module tiling of an embodiment of the invention to illustrate module interconnectivity;

FIG. 2 depicts a module containing a plurality of nodes according to certain embodiments of the invention;

FIG. 3 depicts a module tiling with inferior inter-module connectivity;

FIG. 4 illustrates parallel routing of inter-module signals according to certain embodiments of the invention; and

FIG. 5 depicts data and control links for an inter-node link or connection according to certain embodiments of the invention.

DETAILED DISCUSSION

Preferred embodiments of the invention provide massively parallel computer systems in which processor nodes are interconnected in a Kautz-like topology. Preferred embodiments provide a computing system having O nodes (i.e., order O) equally divided on M modules, each module having N nodes, N=O/M. By appropriately selecting the size N of the module and appropriately selecting the specific set of nodes to be included on a module, the inter-node routing problem may be significantly reduced. Specifically, the inter-node routing may be arranged so that a high percentage of the inter-node connections or links may remain on a module (i.e., intra-module) and avoid inter-module connections, thus reducing the amount of inter-node connections that must involve a backplane, cables, or the like. Moreover, the inter-node connections that must be inter-module (and thus require a backplane or cables, or the like) may be arranged in a parallel fashion. These features facilitate the creation of larger systems and yield inter-node connections with shorter paths and latencies. That is, preferred embodiments provide efficient and effective logical routing (i.e., the number of hops between nodes) and also provide efficient and effective physical routing (i.e., allowing high-speed interconnect to be used on large systems).

Certain embodiments of the invention use a Kautz topology for data links and data flow to interconnect the node, but they are not purely directed graphs because they include a control link back channel link from receiver to sender. This link is used for flow control and status, among other things.

Kautz interconnection topologies are unidirectional, directed graphs (digraphs). Kautz digraphs are characterized by a degree k and a diameter n. The degree of the digraph is the maximum number of arcs (or links or edges) input to or output from any node. The diameter is the maximum number of arcs that must be traversed from any node to any other node in the topology.

The order O of a graph is the number of nodes it contains. The order of a Kautz digraph is (k+1)k^(n−1). The diameter of a Kautz digraph increases logarithmically with the order of the graph.

FIG. 1A depicts a very simple Kautz topology for descriptive convenience. The system 100 has degree three; that is, each node has three ingress links 110 and three egress links 112. The topology has diameter one, meaning that any node can communicate with any other node in a maximum of one hop. The topology is order 4, meaning that there are 4 nodes.

FIG. 1B shows a system that is order 12 and diameter two. By inspection, one can verify that any node can communicate with any other node in a maximum of two hops. FIG. 1C shows a system that is degree three and diameter three, having order 36. One quickly sees that the complexity of the system grows quickly. It would be counter-productive to depict and describe preferred systems such as those having hundreds of nodes or more.

The table below shows how the order O of a system changes as the diameter n grows for a system of fixed degree k.

Order Diameter (n) k = 2 k = 3 k = 4 3 12 36 80 4 24 108 320 5 48 324 1280 6 96 972 5120

With nodes numbered from zero to O-1, the digraph can be constructed by running a link from any node x to any other node y that satisfies the following equation: y=(−x*k−j) mod O, where 1≦j≦k  (1) Thus, any (x,y) pair satisfying (1) specifies a direct egress link from node x. For example, with reference to FIG. 1C node 1 has egress links to the set of nodes 30, 31 and 32. Iterating through this procedure for all nodes in the system will yield the interconnections, links, arcs or edges needed to satisfy the Kautz topology. (As stated above, communication between two arbitrarily selected nodes may require multiple hops through the topology but the number of hops is bounded by the diameter of the topology.)

Under certain embodiments of the invention, the system is arranged into multiple modules. The modules are created to have a particular size (i.e., number of nodes on the module) and a particular set of nodes on the module. It has been observed by the inventors that careful selection of the module size and careful attention to the selection of the set of nodes to included on a given module can significantly reduce wiring problems in systems built with the Kautz topology.

More specifically, under preferred embodiments of the invention, the Kautz topology is uniformly tiled. To do this, the Kautz graph is one-to-one mapped to satisfy the following equation. t: V _(G)→I×V_(T)   (2) In the above, V_(G) is the set of vertices of a Kautz graph; V_(T) is the set of vertices of a tile (i.e., a smaller graph, implemented as a module of nodes); and I is an index set. Moreover, if (x,y) is an edge within tile T then (t⁻¹(i,x), t⁻¹(i,y) is an edge of Kautz graph G.

The tiles or modules are arranged to maximize the number of edges of the tile T. That is, the tiles or modules are arranged so that a maximum number of edges, arc, or links in the Kautz topology are contained on the tiles. All the remaining edges by necessity are inter-tile (or inter-module). By doing this, node interconnections will be maximized to remain intra-module.

Conventionally a Kautz graph of degree k and diameter n can label the vertices of the topology as follows, with each integer s being base k+1. Adjacent integers must differ. s ₁ s ₂ . . . s _(n) ε Z_(k+1) ^(n) , s _(i) ≠s _(i+1)  (3)

A de Bruijn graph is closely related to a Kautz graph. A de Bruijn graph has vertices that may be labeled by strings of n integers base k, as follows: c₁c₂ . . . c_(n) ε Z_(k) ^(n)  (4)

The vertices of a degree k, diameter n Kautz graph can be mapped to the vertices of a degree k, diameter n−1 de Bruijn graph as follows: r: s ₁ . . . s _(n) →c ₁ . . . c _(n−1) , c _(i)=(s _(i+1) −s _(i)) (mod k+1)−1  (5) or any other mapping of s_(i+1) and s_(i) to c_(i) such that with either parameter fixed, the mapping gives unique values of c_(i) mod k for all allowed values of the other parameter.

Consequently, the edges, links or arcs in a Kautz graph may be expressed as follows: (s ₀ c ₁ c ₂ . . . c _(n−1) , [s ₀ +c ₁+1]c ₂ c ₃ . . . c_(n))  (6) where [s₀+c₁+1] is taken modulo k+1.

To make the tiling scalable to arbitrary diameter graphs, the tile M must be equivalent to a subgraph of a de Bruijn graph of diameter m and degree k containing all the nodes of the de Bruijn graph but only a subset of the edges subject to the condition that the edges on the tile cannot form any directed loops. In order to minimize inter-module wiring, the subgraph with the maximal number of intra-module edges (without directed loops) should be chosen subject to the condition that the tile can be extended to form a complete tiling of the system.

To generate a complete tiling, it is possible to use a map Π: G→M from the nodes of the complete graph G to the nodes of the tile M which respects the edge structure of the de Bruijn graph of diameter m on which the tile is based. This map may in particular be chosen to satisfy the following conditions: Π(P(u))−P(Π(u)), ∀u ε G Π(C(u))=C(Π(u)), ∀u ε G where C(u) denotes the set of nodes which are reached from edges beginning at node u and P(u) denotes the set of nodes from which node u can be reached by following a single edge.

Under certain embodiments of the invention, each module has k^(m) nodes, and each node on the module can be assigned a label d₁ . . . d_(m) ε z_(k) ^(m) such that inter-node connections that are intra-module correspond to a subset of the edges (d₁ . . . d_(m), d₂ . . . d_(m+1)) of a de Bruijn graph of diameter m and degree k, subject to the condition that there are no directed closed loops formed from the inter-node connections on a module.

Under certain embodiments of the invention, maps Π satisfying the conditions stated above for P(u) and C(u) may be defined by expressing d_(i)'s as a “discrete differential” function of node labels the s₀ . . . s_(n) of the Kautz graph through d _(i) =f(c _(i+n−m) , c _(i))  (7) wherein f(x,y) is a function which for fixed X acts a permutation on Z_(k) through y→f (X,y) and which for fixed Y acts as a permutation on Z_(k) through x→f(x,Y) and where c_(i)'s encode the Kautz coordinates Si through c _(i) =s _(i) −s _(i−1) mod(k+1)  (8)

Under certain embodiments, f(x,y) equals x+y mod k, or f(x,y) equals x−y mod k.

Given a map Π with the conditions defined above, the tiling may then be defined as follows. Choose a vertex x₀=d₁ . . . d_(N−n) of the tile (or module) T. Associated with this vertex of T is a set of vertices in the larger Kautz graph each of which has the same value of Π(u)=x₀. Define the index set by the remaining indices on this set of vertices (i.e., s₀c₁ . . . c_(n)). This defines t⁻¹(i, x_(o)) for all i. If there are any edges in T containing x_(o) the definition is extended. For example, consider if T contains the edge (x_(o), x₁). For each i in I, there is a unique vertex in the Kautz graph which is reached by an edge from t⁻¹(i, x_(o)) and which has d₁ . . . d_(N−n)=x₁. Define this vertex to be t⁻¹(i, x₁). Continue in the same way for further edges containing either x₀ or x₁ . Each time a new edge is included the map t⁻¹ is defined for the new value of x. In this fashion the complete tiling may be completed.

Tiling constructed in the fashion of the previous discussion automatically have the parallel routing property. The benefits of parallel routing are described below.

FIG. 1D for example shows a module or tile for a very simple Kautz topology of order 36 and degree three. Each module has nine nodes, as depicted.

The table shows how the nodes and modules connect. Notice how the linear labels are distributed among modules. For example, linearly labeled nodes 0-9 are not all assigned to module 0. As mentioned above the interconnection among nodes is defined by equation 1, and the assignment among modules is a result of the tiling method employed. This example of FIG. 1D is particularly simple in comparison to the larger systems of preferred embodiments. The size of preferred embodiments is prohibitively large to depict by figures or tables and instead is explained by the mathematics above. This example is utilized to illustrate the complexity of module assignment and the interconnections among nodes.

Shift Register map to Linear Exits label deBruijn Assignment label 0 1 2 s0 s1 s2 s0 c1 c2 mod node 0 35 34 33 0 3 0 0 2 0 0 6 1 32 31 30 0 3 1 0 2 1 0 7 2 29 28 27 0 3 2 0 2 2 3 8 3 26 25 24 0 2 3 0 1 0 3 3 4 23 22 21 0 2 0 0 1 1 1 4 5 20 19 18 0 2 1 0 1 2 2 5 6 17 16 15 0 1 2 0 0 0 1 0 7 14 13 12 0 1 3 0 0 1 1 1 8 11 10 9 0 1 0 0 0 2 1 2 9 8 7 6 1 0 1 1 2 0 1 6 10 5 4 3 1 0 2 1 2 1 1 7 11 2 1 0 1 0 3 1 2 2 0 8 12 35 34 33 1 3 0 1 1 0 0 3 13 32 31 30 1 3 1 1 1 1 2 4 14 29 28 27 1 3 2 1 1 2 3 5 15 26 25 24 1 2 3 1 0 0 2 0 16 23 22 21 1 2 0 1 0 1 2 1 17 20 19 18 1 2 1 1 0 2 2 2 18 17 16 15 2 1 2 2 2 0 2 6 19 14 13 12 2 1 3 2 2 1 2 7 20 11 10 9 2 1 0 2 2 2 1 8 21 8 7 6 2 0 1 2 1 0 1 3 22 5 4 3 2 0 2 2 1 1 3 4 23 2 1 0 2 0 3 2 1 2 0 5 24 35 34 33 2 3 0 2 0 0 3 0 25 32 31 30 2 3 1 2 0 1 3 1 26 29 28 27 2 3 2 2 0 2 3 2 27 26 25 24 3 2 3 3 2 0 3 6 28 23 22 21 3 2 0 3 2 1 3 7 29 20 19 18 3 2 1 3 2 2 2 8 30 17 16 15 3 1 2 3 1 0 2 3 31 14 13 12 3 1 3 3 1 1 0 4 32 11 10 9 3 1 0 3 1 2 1 5 33 8 7 6 3 0 1 3 0 0 0 0 34 5 4 3 3 0 2 3 0 1 0 1 35 2 1 0 3 0 3 3 0 2 0 2

Under preferred embodiments, module size is an integral power of the degree (k). Certain embodiments maximize this size as described above, i.e., largest subgraph without directed loops, but others may be smaller for practical considerations in building modules. These are substantially optimal in terms of maximizing edges to be intra-module.

Certain embodiments use a module size of 27 nodes where each node is of degree 3. Each module has a particular set of nodes thereon (as described above) and may be used to build Kautz topologies of 108, 324, 972 or more nodes, or de Bruijn topologies with multiples of 27 nodes.

FIG. 2 depicts a module arrangement having 27 nodes, numbered 0 through 26 in the upper right corner of nodes. These node numbers are, in certain embodiments, the numbering schema of equations 7 and 8. That is, the node numbers shown are adjacent in the number space provided by the discrete differential numbering scheme outlined above, though they need not be adjacent in the numbering of nodes of the Kautz topology as expressed in equation 1. The node identifier is expressed in the upper right corner of the node in decimal form, and in the middle of the node it is expressed in ternary form.

As illustrated, each node identifies the egress links 202 and ingress links 204. Focusing on egress links for the time being (with the explanation extending to ingress links too), node 7 has egress links going to nodes 21, 22, and 23 (upper right notation, i.e., node identifier) on other modules in the system. The figure depicts just the numbering scheme and not the node identification within the Kautz topology. As mentioned above, the actual interconnectivity is defined by equation 1. Thus, some connections depicted on FIG. 2 identify node numbers (via its number identifier), which are the same, even though in the larger system the node numbers will go to different nodes. For example, the figure shows nodes 17, 26 and 8, each with output links to another node (off module) identified by number 26. However, the node 26 driven by nodes 17 and 26 (upper right of FIG. 2) is on a different module than the node 26 driven by node 8. The actual nodes involved are governed by the above equations.

FIG. 4 depicts a simplified diagram, drawn in perspective, to illustrate the parallel routing that results from the tiling approach discussed above. A first module 402 has an output pin 404 in communication with backplane trace 408 on backplane 406. (A backplane layer is illustrated, but other structures such as midplanes or the like may be used.) The trace 408 is parallel and horizontal to pin 410 on module 412. That is, the backplane trace has no vertical runs. Under preferred embodiments of the invention, every backplane run will be parallel in a similar manner. Though many layers may be needed for the backplane when there are a significant number of modules, the backplane traces will not need vertical runs to connect the relevant pins and links, and instead runs will be horizontal and parallel. (Alternatively if things were rotated the runs could all be vertical and parallel.) This routing greatly facilitates the ability to keep high signal integrity, which in turn greatly improves the ability to run the inter-node and inter-module connections at very high speed. It also enables larger systems to be built while maintaining satisfactory signal integrity (i.e., designs don't need to decrease bus speed to enable large systems). Using the example of FIG. 2, the trace 408 may correspond to the connection from the node with discrete differential number 5 (lower part of figure) to another node on a different module (412) with discrete differential identifier 17. Notice in the upper right of FIG. 2 that every node 17 receives an input from another node 5 (discrete differential number). In certain embodiments, such as a 972 node system with modules like that shown in FIG. 2, each module will have 39 pins (e.g., 404 and 410), and every backplane trace will run horizontal and parallel to other traces. Only one backplane layer 406 is shown in FIG. 4 for clarity, but a system of 972 nodes may require about 20 such layers. Such a backplane, however, will be faster and have better signal integrity than one that did not have parallel routes and which needed vertical runs, vias and the likes to provide connectivity among modules.

Referring back to FIG. 2, Each node on the system may communicate with any other node on the system by appropriately routing messages onto the communication fabric via an egress link 202. Some of these egress links will be inter-module, such as the ones depicted in connection with node 7. Others will be intra-module, such as those being depicted in connection with node 2 which go to nodes 6, 7, and 8 on the same module. Some nodes have some links intra-module and some inter-module, see for example node 12.

Under certain embodiments, any data message on the fabric includes routing information in the header of the message (among other information). The routing information specifies the entire route of the message. In certain degree three embodiments, the routing information is a bit string of 2-bit routing codes, each routing code specifying whether a message should be received locally (i.e., this is the target node of the message) or identifying one of three egress links. Naturally other topologies may be implemented with different routing codes and with different structures and methods under the principles of the invention. Under certain embodiments, each node has tables programmed with the routing information. For a given node x to communicate with another node z, node x accesses the table and receives a bit string for the routing information. As will be explained below, this bit string is used to control various switches along the message's route to node z, in effect specifying which link to utilize at each node during the route. Another node j may have a different bit string when it needs to communicate with node z, because it will employ a different route to node z and the message may utilize different links at the various nodes in its route to node z. Thus, under certain embodiments, the routing information is not literally an “address” (i.e., it doesn't uniquely identify node z) but instead is a set of codes to control switches for the message's route.

Under certain embodiments, the routes are determined a priori based on the interconnectivity of the Kautz topology as expressed in equation 1. That is, the Kautz topology is defined, and the various egress links for each node are assigned a code (i.e., each link being one of three egress links). Thus, the exact routes for a message from node x to node z are known in advance, and the egress link selections may be determined in advance as well. These link selections are programmed as the routing information. These tables may be reprogrammed as needed, for example, to route around faulty links or nodes.

Certain embodiments modify the routing information in the message header en route for easier processing. For example, a node will analyze a 2 bit field of the routing information to determine which link the message should use, e.g., one of three egress links or it should be kept local (i.e., this is the destination node). This could be the least significant numeral, digits or bits of the routing field, but it need not be limited to such (i.e., it depends on the embodiment). Once a node determines that a message should be forwarded on one of the egress links, the node shifts the routing bit string accordingly (e.g., by 2 bits) so the next node in the route can perform an exactly similar set of operation: i.e., process the lowest two bits of the route code to determine if the message should be handled locally or forwarded on a specific one of three egress links).

The routing information, in these embodiments, is used to identify portions in a cross point buffer to hold the data so that the message may be stored until it may be forwarded on the appropriate link. (Certain embodiments support cut-through routing to avoid the buffer if the appropriate link is not busy when the message arrive or becomes free during reception of the message.)

In certain embodiments, the messages also contain other information such as virtual channel identification information. As explained in more detail in the related and incorporated applications, virtual channel information is used so that each link may be associated with multiple virtual channels and so that deadlock avoidance techniques may be implemented.

Experimentation shows that with a preferred arrangement 48% of the inter-node links may be routed inter-module, and 52% can be routed intra-module. Other degrees, diameters, order, and modules sizes may be used using the principles of the invention.

In contrast, other methods of selecting nodes may yield significantly less intra-connections module connections (and as a result more inter-module connections). FIG. 3 for example shows an arrangement also involving 27 nodes per module. However, even though the arrangement seems well-organized (e.g., tree like) only about 30% of the inter-node connections remains on module, meaning more of the inter-node connections will require a backplane or the like, inhibiting the ability to build larger systems.

Under certain embodiments the computing system is not configured as a Kautz digraph in pure form in that the communication is not purely unidirectional. Instead, certain preferred embodiments have data communication implemented on unidirectional directed links (or circuits) and use a back channel control link (or circuit) for flow control and maintenance purposes.

FIG. 5 for example shows two nodes, sender 502 and receiver 504, following the unidirectional convention used above in discussing Kautz topologies. These nodes could correspond, for example, to two intra-module nodes such as nodes 18 and 2 in FIG. 2. The link 506 connecting the two nodes includes unidirectional data lanes 508 and unidirectional control lanes 510. The direction of the data lanes 508 is consistent with the convention used above in discussing the unidirectional flow of the Kautz digraph. The direction of the control link is in the opposite direction, i.e., from data receiving node 504 to data transmitting node 502. The arrangement is asymmetric in the sense that there are more forward data lane circuits than there are reverse control lane circuits. In certain embodiments there are eight data circuits and one control circuit between two connected nodes.

In certain embodiments each sender 502 assigns a link sequence number (LSN) to every outgoing packet. The LSN is included in the packet header. The sender 502 also keeps transmitted packets in a replay buffer until it has been confirmed (more below) that the packets have been successfully received.

Receiver nodes receive packets and keep track of the LSN of the most recently received error free packet as part of its buffer status. Periodically, the receiver node 504 transmits buffer status back to the sender using the control circuit 510. In certain embodiments, this status is transmitted as frequently as possible. The LSN corresponds to the most recently received packet if there has been no error. If there has been an error detected, the buffer status will indicate error and include the LSN of the last packet correctly received.

In response the sending node 502 identifies the LSN in the buffer status packet and from this realizes that all packets up to and including the identified LSN have been received at the receiving node 504 in acceptable condition. The sender 502 may then delete packets from the replay buffer with LSNs up to and including the LSN received in the status packet. If an error has been detected, the sender will resend all packets in the replay buffer starting after the LSN of the buffer status (the receiving node will have dropped such in anticipation of the replay and to ensure that all packets from the same source, going to the same destination, along the same route, with the same virtual channel are delivered and kept in order). Thus, packet error detection and recovery is performed at the link level. Likewise packets are guaranteed to be delivered in order at the link-level.

The control circuits are also used to convey buffer status information for downstream nodes to indicate whether buffer space associated with virtual channels are free or busy. As is explained in the incorporated patent applications, the nodes use a cross point buffer to store data from the links and to organize and control the data flow as virtual channel assignments over the links to avoid deadlock. More specifically, a debit/credit mechanism is used in which the receiving node 504 informs the sending node 502 of how much space is available in the buffers (not shown) of the receiving node 504 for each virtual channel and port. Under certain embodiments a sender 502 will not send information unless it knows that there is buffer space for the virtual channel in the next downstream node along the route. The control packet stream carries a current snapshot of the cross point buffer entry utilization for each of the crosspoint buffers it has (which depends on the degree of the system).

The control link may also be used for out-of-band communication between connected nodes by using otherwise unused fields in the packet to communicate. This mechanism may be used for miscellaneous purposes.

In a Kautz network no single or (if degree three or higher) double failure can isolate any working node or subset of nodes from the rest of the network. No single link or node failure increases the network diameter by more than one hop. Certain embodiments of the invention use multiple paths in the topology to avoid congestion and faulty links or nodes.

Many of the teachings here may be extended to other topologies including de Bruijn topologies. Likewise, though the description was in relation to large-scale computing system, the principles may apply to other digital systems.

Certain embodiments used discrete differential in the low order positions of the label identification. This is particularly helpful for parallel routing.

The above discussion concerning Kautz tilings are applicable to de Bruijn topologies as well.

Certain embodiments of the invention allow what are above described as a tile to be combined on to module. For example, two tiles may be formed on a module, and a module under these arrangement will have pk^(m) nodes where p is an integer.

Appendix A (attached) is a listing of a particular 972 node, 36 module, degree three system. The columns identify the Kautz number (0-971), the node identification (per module) and specify the other nodes to which each node connects. From this, one can determine node-to-node interconnectivity for each node in the system.

While the invention has been described in connection with certain preferred embodiments, it will be understood that it is not intended to limit the invention to those particular embodiments. On the contrary, it is intended to cover all alternatives, modifications and equivalents as may be included in the appended claims. Some specific figures and source code languages are mentioned, but it is to be understood that such figures and languages are, however, given as examples only and are not intended to limit the scope of this invention in any manner.

APPENDIX A S Bd Md Nd Kid FM1 Fm2 Fm3 To1 To2 To3 0 0 0 0 168 267 915 591 465 466 467 0 0 0 1 169 267 915 591 462 463 464 0 0 0 2 267 558 234 882 170 168 169 0 0 0 3 493 807 159 483 462 463 464 0 0 0 4 492 807 159 483 465 466 467 0 0 0 5 817 51 699 375 462 463 464 0 0 0 6 816 51 699 375 465 466 467 0 0 0 7 462 169 493 817 556 557 555 0 0 0 8 463 169 493 817 553 554 552 0 0 0 9 464 169 493 817 549 551 550 0 0 0 10 465 168 492 816 548 547 546 0 0 0 11 466 168 492 816 545 544 543 0 0 0 12 467 168 492 816 541 540 542 0 0 0 13 915 18 666 342 170 168 169 0 0 0 14 591 774 126 450 170 168 169 0 0 0 15 159 918 270 594 494 493 492 0 0 0 16 161 918 270 594 488 487 486 0 0 0 17 483 162 810 486 494 493 492 0 0 0 18 699 90 738 414 818 817 816 0 0 0 19 700 90 738 414 815 813 814 0 0 0 20 375 846 198 522 818 817 816 0 0 0 21 270 881 233 557 160 159 161 0 0 0 22 594 125 773 449 160 159 161 0 0 0 23 738 77 725 401 701 699 700 0 0 0 24 414 509 833 185 701 699 700 0 0 0 25 486 809 485 161 484 485 483 0 0 0 26 522 149 473 797 377 376 375 0 1 1 0 98 291 939 615 675 676 677 0 1 1 1 96 291 939 615 681 682 683 0 1 1 2 291 874 550 226 97 98 96 0 1 1 3 420 507 831 183 681 682 683 0 1 1 4 422 507 831 183 675 676 677 0 1 1 5 744 75 723 399 681 682 683 0 1 1 6 746 75 723 399 675 676 677 0 1 1 7 681 96 420 744 872 870 871 0 1 1 8 682 96 420 744 869 867 868 0 1 1 9 683 96 420 744 865 864 866 0 1 1 10 675 98 422 746 888 890 889 0 1 1 11 676 98 422 746 885 887 886 0 1 1 12 677 98 422 746 884 883 882 0 1 1 13 939 334 10 658 97 98 96 0 1 1 14 615 118 442 766 97 98 96 0 1 1 15 831 46 370 694 421 420 422 0 1 1 16 833 46 370 694 415 414 416 0 1 1 17 183 262 910 586 421 420 422 0 1 1 18 723 406 82 730 745 744 746 0 1 1 19 724 406 82 730 742 743 741 0 1 1 20 399 190 514 838 745 744 746 0 1 1 21 370 848 200 524 832 831 833 0 1 1 22 694 92 740 416 832 831 833 0 1 1 23 82 296 944 620 725 723 724 0 1 1 24 730 80 404 728 725 723 724 0 1 1 25 586 776 452 128 184 185 183 0 1 1 26 838 368 692 44 401 400 399 0 2 2 0 25 315 963 639 894 895 896 0 2 2 1 26 315 963 639 891 892 893 0 2 2 2 315 218 866 542 24 25 26 0 2 2 3 350 531 855 207 891 892 893 0 2 2 4 349 531 855 207 894 895 896 0 2 2 5 674 747 423 99 891 892 893 0 2 2 6 673 747 423 99 894 895 896 0 2 2 7 891 26 350 674 240 241 242 0 2 2 8 892 26 350 674 237 238 239 0 2 2 9 893 26 350 674 236 235 234 0 2 2 10 894 25 349 673 232 231 233 0 2 2 11 895 25 349 673 229 228 230 0 2 2 12 896 25 349 673 225 227 226 0 2 2 13 963 650 326 2 24 25 26 0 2 2 14 639 434 758 110 24 25 26 0 2 2 15 855 362 686 38 348 350 349 0 2 2 16 857 362 686 38 342 344 343 0 2 2 17 207 578 254 902 348 350 349 0 2 2 18 423 506 182 830 672 674 673 0 2 2 19 424 506 182 830 669 670 671 0 2 2 20 99 290 614 938 672 674 673 0 2 2 21 686 95 419 743 856 855 857 0 2 2 22 38 635 311 959 856 855 857 0 2 2 23 182 263 911 587 425 423 424 0 2 2 24 830 47 371 695 425 423 424 0 2 2 25 902 23 671 347 208 209 207 0 2 2 26 938 335 659 11 101 100 99 0 3 3 0 192 583 259 907 395 393 394 0 3 3 1 193 583 259 907 392 390 391 0 3 3 2 583 777 453 129 194 192 193 0 3 3 3 517 151 475 799 392 390 391 0 3 3 4 516 151 475 799 395 393 394 0 3 3 5 841 367 43 691 392 390 391 0 3 3 6 840 367 43 691 395 393 394 0 3 3 7 392 193 517 841 766 767 765 0 3 3 8 390 193 517 841 772 773 771 0 3 3 9 391 193 517 841 768 770 769 0 3 3 10 395 192 516 840 758 757 756 0 3 3 11 393 192 516 840 764 763 762 0 3 3 12 394 192 516 840 760 759 761 0 3 3 13 259 561 237 885 194 192 193 0 3 3 14 907 21 345 669 194 192 193 0 3 3 15 475 165 489 813 518 517 516 0 3 3 16 474 165 489 813 521 520 519 0 3 3 17 799 705 381 57 518 517 516 0 3 3 18 43 633 309 957 842 841 840 0 3 3 19 44 633 309 957 839 837 838 0 3 3 20 691 93 417 741 842 841 840 0 3 3 21 489 808 160 484 476 475 474 0 3 3 22 813 52 700 376 476 475 474 0 3 3 23 309 220 868 544 42 43 44 0 3 3 24 957 652 4 328 42 43 44 0 3 3 25 57 952 628 304 800 798 799 0 3 3 26 741 76 400 724 690 692 691 0 4 4 0 122 607 283 931 605 603 604 0 4 4 1 120 607 283 931 611 609 610 0 4 4 2 607 121 769 445 121 122 120 0 4 4 3 444 823 175 499 611 609 610 0 4 4 4 446 823 175 499 605 603 604 0 4 4 5 768 391 67 715 611 609 610 0 4 4 6 770 391 67 715 605 603 604 0 4 4 7 611 120 444 768 110 108 109 0 4 4 8 609 120 444 768 116 114 115 0 4 4 9 610 120 444 768 112 111 113 0 4 4 10 605 122 446 770 126 128 127 0 4 4 11 603 122 446 770 132 134 133 0 4 4 12 604 122 446 770 131 130 129 0 4 4 13 283 877 553 229 121 122 120 0 4 4 14 931 337 661 13 121 122 120 0 4 4 15 175 265 589 913 445 444 446 0 4 4 16 174 265 589 913 448 447 449 0 4 4 17 499 805 481 157 445 444 446 0 4 4 18 67 949 625 301 769 768 770 0 4 4 19 68 949 625 301 766 767 765 0 4 4 20 715 409 733 85 769 768 770 0 4 4 21 589 775 127 451 176 175 174 0 4 4 22 913 19 667 343 176 175 174 0 4 4 23 625 439 115 763 66 67 68 0 4 4 24 301 223 547 871 66 67 68 0 4 4 25 157 919 595 271 500 498 499 0 4 4 26 85 295 619 943 714 716 715 0 5 5 0 49 631 307 955 824 822 823 0 5 5 1 50 631 307 955 821 819 820 0 5 5 2 631 437 113 761 48 49 50 0 5 5 3 374 847 199 523 821 819 820 0 5 5 4 373 847 199 523 824 822 823 0 5 5 5 698 91 739 415 821 819 820 0 5 5 6 697 91 739 415 824 822 823 0 5 5 7 821 50 374 698 450 451 452 0 5 5 8 819 50 374 698 456 457 458 0 5 5 9 820 50 374 698 455 454 453 0 5 5 10 824 49 373 697 442 441 443 0 5 5 11 822 49 373 697 448 447 449 0 5 5 12 823 49 373 697 444 446 445 0 5 5 13 307 221 869 545 48 49 50 0 5 5 14 955 653 5 329 48 49 50 0 5 5 15 199 581 905 257 372 374 373 0 5 5 16 198 581 905 257 375 377 376 0 5 5 17 523 149 797 473 372 374 373 0 5 5 18 739 77 725 401 696 698 697 0 5 5 19 740 77 725 401 693 694 695 0 5 5 20 415 509 833 185 696 698 697 0 5 5 21 905 22 346 670 200 199 198 0 5 5 22 257 562 238 886 200 199 198 0 5 5 23 725 406 82 730 738 739 740 0 5 5 24 401 190 514 838 738 739 740 0 5 5 25 473 166 814 490 524 522 523 0 5 5 26 185 262 586 910 414 416 415 0 6 6 0 216 899 575 251 322 323 321 0 6 6 1 217 899 575 251 319 320 318 0 6 6 2 899 24 672 348 218 216 217 0 6 6 3 541 467 791 143 319 320 318 0 6 6 4 540 467 791 143 322 323 321 0 6 6 5 865 683 359 35 319 320 318 0 6 6 6 864 683 359 35 322 323 321 0 6 6 7 319 217 541 865 13 14 12 0 6 6 8 320 217 541 865 10 11 9 0 6 6 9 318 217 541 865 15 17 16 0 6 6 10 322 216 540 864 5 4 3 0 6 6 11 323 216 540 864 2 1 0 0 6 6 12 321 216 540 864 7 6 8 0 6 6 13 575 780 456 132 218 216 217 0 6 6 14 251 564 888 240 218 216 217 0 6 6 15 791 708 60 384 542 541 540 0 6 6 16 790 708 60 384 545 544 543 0 6 6 17 143 924 600 276 542 541 540 0 6 6 18 359 852 528 204 866 865 864 0 6 6 19 357 852 528 204 872 870 871 0 6 6 20 35 636 960 312 866 865 864 0 6 6 21 60 951 303 627 789 791 790 0 6 6 22 384 195 843 519 789 791 790 0 6 6 23 528 147 795 471 358 359 357 0 6 6 24 204 579 903 255 358 359 357 0 6 6 25 276 879 555 231 141 142 143 0 6 6 26 312 219 543 867 34 33 35 0 7 7 0 146 923 599 275 532 533 531 0 7 7 1 144 923 599 275 538 539 537 0 7 7 2 923 340 16 664 145 146 144 0 7 7 3 468 167 491 815 538 539 537 0 7 7 4 470 167 491 815 532 533 531 0 7 7 5 792 707 383 59 538 539 537 0 7 7 6 794 707 383 59 532 533 531 0 7 7 7 538 144 468 792 329 327 328 0 7 7 8 539 144 468 792 326 324 325 0 7 7 9 537 144 468 792 331 330 332 0 7 7 10 532 146 470 794 345 347 346 0 7 7 11 533 146 470 794 342 344 343 0 7 7 12 531 146 470 794 350 349 348 0 7 7 13 599 124 772 448 145 146 144 0 7 7 14 275 880 232 556 145 146 144 0 7 7 15 491 808 160 484 469 468 470 0 7 7 16 490 808 160 484 472 471 473 0 7 7 17 815 52 700 376 469 468 470 0 7 7 18 383 196 844 520 793 792 794 0 7 7 19 381 196 844 520 799 800 798 0 7 7 20 59 952 304 628 793 792 794 0 7 7 21 160 918 270 594 489 491 490 0 7 7 22 484 162 810 486 489 491 490 0 7 7 23 844 366 42 690 382 383 381 0 7 7 24 520 150 474 798 382 383 381 0 7 7 25 376 846 522 198 813 814 815 0 7 7 26 628 438 762 114 58 57 59 0 8 8 0 73 947 623 299 751 752 750 0 8 8 1 74 947 623 299 748 749 747 0 8 8 2 947 656 332 8 72 73 74 0 8 8 3 398 191 515 839 748 749 747 0 8 8 4 397 191 515 839 751 752 750 0 8 8 5 722 407 83 731 748 749 747 0 8 8 6 721 407 83 731 751 752 750 0 8 8 7 748 74 398 722 669 670 671 0 8 8 8 749 74 398 722 666 667 668 0 8 8 9 747 74 398 722 674 673 672 0 8 8 10 751 73 397 721 661 660 662 0 8 8 11 752 73 397 721 658 657 659 0 8 8 12 750 73 397 721 663 665 664 0 8 8 13 623 440 116 764 72 73 74 0 8 8 14 299 224 548 872 72 73 74 0 8 8 15 515 152 476 800 396 398 397 0 8 8 16 514 152 476 800 399 401 400 0 8 8 17 839 368 44 692 396 398 397 0 8 8 18 83 296 944 620 720 722 721 0 8 8 19 81 296 944 620 726 727 728 0 8 8 20 731 80 404 728 720 722 721 0 8 8 21 476 165 489 813 513 515 514 0 8 8 22 800 705 381 57 513 515 514 0 8 8 23 944 333 9 657 82 83 81 0 8 8 24 620 117 441 765 82 83 81 0 8 8 25 692 93 741 417 837 838 839 0 8 8 26 728 405 729 81 730 729 731 1 0 9 0 411 510 186 834 708 709 710 1 0 9 1 412 510 186 834 705 706 707 1 0 9 2 510 801 477 153 413 411 412 1 0 9 3 736 78 402 726 705 706 707 1 0 9 4 735 78 402 726 708 709 710 1 0 9 5 88 294 942 618 705 706 707 1 0 9 6 87 294 942 618 708 709 710 1 0 9 7 705 412 736 88 799 800 798 1 0 9 8 706 412 736 88 796 797 795 1 0 9 9 707 412 736 88 792 794 793 1 0 9 10 708 411 735 87 791 790 789 1 0 9 11 709 411 735 87 788 787 786 1 0 9 12 710 411 735 87 784 783 785 1 0 9 13 186 261 909 585 413 411 412 1 0 9 14 834 45 369 693 413 411 412 1 0 9 15 402 189 513 837 737 736 735 1 0 9 16 404 189 513 837 731 730 729 1 0 9 17 726 405 81 729 737 736 735 1 0 9 18 942 333 9 657 89 88 87 1 0 9 19 943 333 9 657 86 84 85 1 0 9 20 618 117 441 765 89 88 87 1 0 9 21 513 152 476 800 403 402 404 1 0 9 22 837 368 44 692 403 402 404 1 0 9 23 9 320 968 644 944 942 943 1 0 9 24 657 752 104 428 944 942 943 1 0 9 25 729 80 728 404 727 728 726 1 0 9 26 765 392 716 68 620 619 618 1 1 10 0 341 534 210 858 918 919 920 1 1 10 1 339 534 210 858 924 925 926 1 1 10 2 534 145 793 469 340 341 339 1 1 10 3 663 750 102 426 924 925 926 1 1 10 4 665 750 102 426 918 919 920 1 1 10 5 15 318 966 642 924 925 926 1 1 10 6 17 318 966 642 918 919 920 1 1 10 7 924 339 663 15 143 141 142 1 1 10 8 925 339 663 15 140 138 139 1 1 10 9 926 339 663 15 136 135 137 1 1 10 10 918 341 665 17 159 161 160 1 1 10 11 919 341 665 17 156 158 157 1 1 10 12 920 341 665 17 155 154 153 1 1 10 13 210 577 253 901 340 341 339 1 1 10 14 858 361 685 37 340 341 339 1 1 10 15 102 289 613 937 664 663 665 1 1 10 16 104 289 613 937 658 657 659 1 1 10 17 426 505 181 829 664 663 665 1 1 10 18 966 649 325 1 16 15 17 1 1 10 19 967 649 325 1 13 14 12 1 1 10 20 642 433 757 109 16 15 17 1 1 10 21 613 119 443 767 103 102 104 1 1 10 22 937 335 11 659 103 102 104 1 1 10 23 325 539 215 863 968 966 967 1 1 10 24 1 323 647 971 968 966 967 1 1 10 25 829 47 695 371 427 428 426 1 1 10 26 109 611 935 287 644 643 642 1 2 11 0 268 558 234 882 165 166 167 1 2 11 1 269 558 234 882 162 163 164 1 2 11 2 558 461 137 785 267 268 269 1 2 11 3 593 774 126 450 162 163 164 1 2 11 4 592 774 126 450 165 166 167 1 2 11 5 917 18 666 342 162 163 164 1 2 11 6 916 18 666 342 165 166 167 1 2 11 7 162 269 593 917 483 484 485 1 2 11 8 163 269 593 917 480 481 482 1 2 11 9 164 269 593 917 479 478 477 1 2 11 10 165 268 592 916 475 474 476 1 2 11 11 166 268 592 916 472 471 473 1 2 11 12 167 268 592 916 468 470 469 1 2 11 13 234 893 569 245 267 268 269 1 2 11 14 882 677 29 353 267 268 269 1 2 11 15 126 605 929 281 591 593 592 1 2 11 16 128 605 929 281 585 587 586 1 2 11 17 450 821 497 173 591 593 592 1 2 11 18 666 749 425 101 915 917 916 1 2 11 19 667 749 425 101 912 913 914 1 2 11 20 342 533 857 209 915 917 916 1 2 11 21 929 338 662 14 127 126 128 1 2 11 22 281 878 554 230 127 126 128 1 2 11 23 425 506 182 830 668 666 667 1 2 11 24 101 290 614 938 668 666 667 1 2 11 25 173 266 914 590 451 452 450 1 2 11 26 209 578 902 254 344 343 342 1 3 12 0 435 826 502 178 638 636 637 1 3 12 1 436 826 502 178 635 633 634 1 3 12 2 826 48 696 372 437 435 436 1 3 12 3 760 394 718 70 635 633 634 1 3 12 4 759 394 718 70 638 636 637 1 3 12 5 112 610 286 934 635 633 634 1 3 12 6 111 610 286 934 638 636 637 1 3 12 7 635 436 760 112 37 38 36 1 3 12 8 633 436 760 112 43 44 42 1 3 12 9 634 436 760 112 39 41 40 1 3 12 10 638 435 759 111 29 28 27 1 3 12 11 636 435 759 111 35 34 33 1 3 12 12 637 435 759 111 31 30 32 1 3 12 13 502 804 480 156 437 435 436 1 3 12 14 178 264 588 912 437 435 436 1 3 12 15 718 408 732 84 761 760 759 1 3 12 16 717 408 732 84 764 763 762 1 3 12 17 70 948 624 300 761 760 759 1 3 12 18 286 876 552 228 113 112 111 1 3 12 19 287 876 552 228 110 108 109 1 3 12 20 934 336 660 12 113 112 111 1 3 12 21 732 79 403 727 719 718 717 1 3 12 22 84 295 943 619 719 718 717 1 3 12 23 552 463 139 787 285 286 287 1 3 12 24 228 895 247 571 285 286 287 1 3 12 25 300 223 871 547 71 69 70 1 3 12 26 12 319 643 967 933 935 934 1 4 13 0 365 850 526 202 848 846 847 1 4 13 1 363 850 526 202 854 852 853 1 4 13 2 850 364 40 688 364 365 363 1 4 13 3 687 94 418 742 854 852 853 1 4 13 4 689 94 418 742 848 846 847 1 4 13 5 39 634 310 958 854 852 853 1 4 13 6 41 634 310 958 848 846 847 1 4 13 7 854 363 687 39 353 351 352 1 4 13 8 852 363 687 39 359 357 358 1 4 13 9 853 363 687 39 355 354 356 1 4 13 10 848 365 689 41 369 371 370 1 4 13 11 846 365 689 41 375 377 376 1 4 13 12 847 365 689 41 374 373 372 1 4 13 13 526 148 796 472 364 365 363 1 4 13 14 202 580 904 256 364 365 363 1 4 13 15 418 508 832 184 688 687 689 1 4 13 16 417 508 832 184 691 690 692 1 4 13 17 742 76 724 400 688 687 689 1 4 13 18 310 220 868 544 40 39 41 1 4 13 19 311 220 868 544 37 38 36 1 4 13 20 958 652 4 328 40 39 41 1 4 13 21 832 46 370 694 419 418 417 1 4 13 22 184 262 910 586 419 418 417 1 4 13 23 868 682 358 34 309 310 311 1 4 13 24 544 466 790 142 309 310 311 1 4 13 25 400 190 838 514 743 741 742 1 4 13 26 328 538 862 214 957 959 958 1 5 14 0 292 874 550 226 95 93 94 1 5 14 1 293 874 550 226 92 90 91 1 5 14 2 874 680 356 32 291 292 293 1 5 14 3 617 118 442 766 92 90 91 1 5 14 4 616 118 442 766 95 93 94 1 5 14 5 941 334 10 658 92 90 91 1 5 14 6 940 334 10 658 95 93 94 1 5 14 7 92 293 617 941 693 694 695 1 5 14 8 90 293 617 941 699 700 701 1 5 14 9 91 293 617 941 698 697 696 1 5 14 10 95 292 616 940 685 684 686 1 5 14 11 93 292 616 940 691 690 692 1 5 14 12 94 292 616 940 687 689 688 1 5 14 13 550 464 140 788 291 292 293 1 5 14 14 226 896 248 572 291 292 293 1 5 14 15 442 824 176 500 615 617 616 1 5 14 16 441 824 176 500 618 620 619 1 5 14 17 766 392 68 716 615 617 616 1 5 14 18 10 320 968 644 939 941 940 1 5 14 19 11 320 368 644 936 937 938 1 5 14 20 658 752 104 428 939 941 940 1 5 14 21 176 265 589 913 443 442 441 1 5 14 22 500 805 481 157 443 442 441 1 5 14 23 968 649 325 1 9 10 11 1 5 14 24 644 433 757 109 9 10 11 1 5 14 25 716 409 85 733 767 765 766 1 5 14 26 428 505 829 181 657 659 658 1 6 15 0 459 170 818 494 565 566 564 1 6 15 1 460 170 818 494 562 563 561 1 6 15 2 170 267 915 591 461 459 460 1 6 15 3 784 710 62 386 562 563 561 1 6 15 4 783 710 62 386 565 566 564 1 6 15 5 136 926 602 278 562 563 561 1 6 15 6 135 926 602 278 565 566 564 1 6 15 7 562 460 784 136 256 257 255 1 6 15 8 563 460 784 136 253 254 252 1 6 15 9 561 460 784 136 258 260 259 1 6 15 10 565 459 783 135 248 247 246 1 6 15 11 566 459 783 135 245 244 243 1 6 15 12 564 459 783 135 250 249 251 1 6 15 13 818 51 699 375 461 459 460 1 6 15 14 494 807 159 483 461 459 460 1 6 15 15 62 951 303 627 785 784 783 1 6 15 16 61 951 303 627 788 787 786 1 6 15 17 386 195 843 519 785 784 783 1 6 15 18 602 123 771 447 137 136 135 1 6 15 19 600 123 771 447 143 141 142 1 6 15 20 278 879 231 555 137 136 135 1 6 15 21 303 222 546 870 60 62 61 1 6 15 22 627 438 114 762 60 62 61 1 6 15 23 771 390 66 714 601 602 600 1 6 15 24 447 822 174 498 601 602 600 1 6 15 25 519 150 798 474 384 385 386 1 6 15 26 555 462 786 138 277 276 278 1 7 16 0 389 194 842 518 775 776 774 1 7 16 1 387 194 842 518 781 782 780 1 7 16 2 194 583 259 907 388 389 387 1 7 16 3 711 410 734 86 781 782 780 1 7 16 4 713 410 734 86 775 776 774 1 7 16 5 63 950 626 302 781 782 780 1 7 16 6 65 950 626 302 775 776 774 1 7 16 7 781 387 711 63 572 570 571 1 7 16 8 782 387 711 63 569 567 568 1 7 16 9 780 387 711 63 574 573 575 1 7 16 10 775 389 713 65 588 590 589 1 7 16 11 776 389 713 65 585 587 586 1 7 16 12 774 389 713 65 593 592 591 1 7 16 13 842 367 43 691 388 389 387 1 7 16 14 518 151 475 799 388 389 387 1 7 16 15 734 79 403 727 712 711 713 1 7 16 16 733 79 403 727 715 714 716 1 7 16 17 86 295 943 619 712 711 713 1 7 16 18 626 439 115 763 64 63 65 1 7 16 19 624 439 115 763 70 71 69 1 7 16 20 302 223 547 871 64 63 65 1 7 16 21 403 189 513 837 732 734 733 1 7 16 22 727 405 81 729 732 734 733 1 7 16 23 115 609 285 933 625 626 624 1 7 16 24 763 393 717 69 625 626 624 1 7 16 25 619 117 765 441 84 85 86 1 7 16 26 871 681 33 357 301 300 302 1 8 17 0 316 218 866 542 22 23 21 1 8 17 1 317 218 866 542 19 20 18 1 8 17 2 218 899 575 251 315 316 317 1 8 17 3 641 434 758 110 19 20 18 1 8 17 4 640 434 758 110 22 23 21 1 8 17 5 965 650 326 2 19 20 18 1 8 17 6 964 650 326 2 22 23 21 1 8 17 7 19 317 641 965 912 913 914 1 8 17 8 20 317 641 965 909 910 911 1 8 17 9 18 317 641 965 917 916 915 1 8 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61 60 2 0 18 11 952 654 6 330 59 58 57 2 0 18 12 953 654 6 330 55 54 56 2 0 18 13 429 504 180 828 656 654 655 2 0 18 14 105 288 612 936 656 654 655 2 0 18 15 645 432 756 108 8 7 6 2 0 18 16 647 432 756 108 2 1 0 2 0 18 17 969 648 324 0 8 7 6 2 0 18 18 213 576 252 900 332 331 330 2 0 18 19 214 576 252 900 329 327 328 2 0 18 20 861 360 684 36 332 331 330 2 0 18 21 756 395 719 71 646 645 647 2 0 18 22 108 611 287 935 646 645 647 2 0 18 23 252 563 239 887 215 213 214 2 0 18 24 900 23 347 671 215 213 214 2 0 18 25 0 323 971 647 970 971 969 2 0 18 26 36 635 959 311 863 862 861 2 1 19 0 584 777 453 129 189 190 191 2 1 19 1 582 777 453 129 195 196 197 2 1 19 2 777 388 64 712 583 584 582 2 1 19 3 906 21 345 669 195 196 197 2 1 19 4 908 21 345 669 189 190 191 2 1 19 5 258 561 237 885 195 196 197 2 1 19 6 260 561 237 885 189 190 191 2 1 19 7 195 582 906 258 386 384 385 2 1 19 8 196 582 906 258 383 381 382 2 1 19 9 197 582 906 258 379 378 380 2 1 19 10 189 584 908 260 402 404 403 2 1 19 11 190 584 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11 409 511 835 187 715 714 716 2 2 20 12 410 511 835 187 711 713 712 2 2 20 13 477 164 812 488 510 511 512 2 2 20 14 153 920 272 596 510 511 512 2 2 20 15 369 848 200 524 834 836 835 2 2 20 16 371 848 200 524 828 830 829 2 2 20 17 693 92 740 416 834 836 835 2 2 20 18 909 20 668 344 186 188 187 2 2 20 19 910 20 668 344 183 184 185 2 2 20 20 585 776 128 452 186 188 187 2 2 20 21 200 581 905 257 370 369 371 2 2 20 22 524 149 797 473 370 369 371 2 2 20 23 668 749 425 101 911 909 910 2 2 20 24 344 533 857 209 911 909 910 2 2 20 25 416 509 185 833 694 695 693 2 2 20 26 452 821 173 497 587 586 585 2 3 21 0 678 97 745 421 881 879 880 2 3 21 1 679 97 745 421 878 876 877 2 3 21 2 97 291 939 615 680 678 679 2 3 21 3 31 637 961 313 878 876 877 2 3 21 4 30 637 961 313 881 879 880 2 3 21 5 355 853 529 205 878 876 877 2 3 21 6 354 853 529 205 881 879 880 2 3 21 7 878 679 31 355 280 281 279 2 3 21 8 876 679 31 355 286 287 285 2 3 21 9 877 679 31 355 282 284 283 2 3 21 10 881 678 30 354 272 271 270 2 3 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5 23 11 336 535 859 211 934 933 935 2 5 23 12 337 535 859 211 930 932 931 2 5 23 13 793 707 383 59 534 535 536 2 5 23 14 469 167 491 815 534 535 536 2 5 23 15 685 95 419 743 858 860 859 2 5 23 16 684 95 419 743 861 863 862 2 5 23 17 37 635 311 959 858 860 859 2 5 23 18 253 563 239 887 210 212 211 2 5 23 19 254 563 239 887 207 208 209 2 5 23 20 901 23 347 671 210 212 211 2 5 23 21 419 508 832 184 686 685 684 2 5 23 22 743 76 724 400 686 685 684 2 5 23 23 239 892 568 244 252 253 254 2 5 23 24 887 676 28 352 252 253 254 2 5 23 25 959 652 328 4 38 36 37 2 5 23 26 671 748 100 424 900 902 901 2 6 24 0 702 413 89 737 808 809 807 2 6 24 1 703 413 89 737 805 806 804 2 6 24 2 413 510 186 834 704 702 703 2 6 24 3 55 953 305 629 805 806 804 2 6 24 4 54 953 305 629 808 809 807 2 6 24 5 379 197 845 521 805 806 804 2 6 24 6 378 197 845 521 808 809 807 2 6 24 7 805 703 55 379 499 500 498 2 6 24 8 806 703 55 379 496 497 495 2 6 24 9 804 703 55 379 501 503 502 2 6 24 10 808 702 54 378 491 490 489 2 6 24 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12 224 897 249 573 298 297 299 3 0 27 13 672 747 423 99 899 897 898 3 0 27 14 348 531 855 207 899 897 898 3 0 27 15 888 675 27 351 251 250 249 3 0 27 16 890 675 27 351 245 244 243 3 0 27 17 240 891 567 243 251 250 249 3 0 27 18 456 819 495 171 575 574 573 3 0 27 19 457 819 495 737 572 570 571 3 0 27 20 132 603 927 279 575 574 573 3 0 27 21 27 638 962 314 889 888 890 3 0 27 22 351 854 530 206 889 888 890 3 0 27 23 495 806 482 158 458 456 457 3 0 27 24 171 266 590 914 458 456 457 3 0 27 25 243 566 242 890 241 242 240 3 0 27 26 279 878 230 554 134 133 132 3 1 28 0 827 48 696 372 432 433 434 3 1 28 1 825 48 696 372 438 439 440 3 1 28 2 48 631 307 955 826 827 825 3 1 28 3 177 264 588 912 438 439 440 3 1 28 4 179 264 588 912 432 433 434 3 1 28 5 501 804 480 156 438 439 440 3 1 28 6 503 804 480 156 432 433 434 3 1 28 7 438 825 177 501 629 627 628 3 1 28 8 439 825 177 501 626 624 625 3 1 28 9 440 825 177 501 622 621 623 3 1 28 10 432 827 179 503 645 647 646 3 1 28 11 433 827 179 503 642 644 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5 32 11 579 778 130 454 205 204 206 3 5 32 12 580 778 130 454 201 203 202 3 5 32 13 64 950 626 302 777 778 779 3 5 32 14 712 410 734 86 777 778 779 3 5 32 15 928 338 662 14 129 131 130 3 5 32 16 927 338 662 14 132 134 133 3 5 32 17 280 878 554 230 129 131 130 3 5 32 18 496 806 482 158 453 455 454 3 5 32 19 497 806 482 158 450 451 452 3 5 32 20 172 266 590 914 453 455 454 3 5 32 21 662 751 103 427 929 928 927 3 5 32 22 14 319 967 643 929 928 927 3 5 32 23 482 163 811 487 495 496 497 3 5 32 24 158 919 271 595 495 496 497 3 5 32 25 230 895 571 247 281 279 280 3 5 32 26 914 19 343 667 171 173 172 3 6 33 0 945 656 332 8 79 80 78 3 6 33 1 946 656 332 8 76 77 75 3 6 33 2 656 753 429 105 947 945 946 3 6 33 3 298 224 548 872 76 77 75 3 6 33 4 297 224 548 872 79 80 78 3 6 33 5 622 440 116 764 76 77 75 3 6 33 6 621 440 116 764 79 80 78 3 6 33 7 76 946 298 622 742 743 741 3 6 33 8 77 946 298 622 739 740 738 3 6 33 9 75 946 298 622 744 746 745 3 6 33 10 79 945 297 621 734 733 732 3 6 33 11 80 945 297 621 731 730 729 3 6 33 12 78 945 297 621 736 735 737 3 6 33 13 332 537 213 861 947 945 946 3 6 33 14 8 321 645 969 947 945 946 3 6 33 15 548 465 789 141 299 298 297 3 6 33 16 547 465 789 141 302 301 300 3 6 33 17 872 681 357 33 299 298 297 3 6 33 18 116 609 285 933 623 622 621 3 6 33 19 114 609 285 933 629 627 628 3 6 33 20 764 393 717 69 623 622 621 3 6 33 21 789 708 60 384 546 548 547 3 6 33 22 141 924 600 276 546 548 547 3 6 33 23 285 876 552 228 115 116 114 3 6 33 24 933 336 660 12 115 116 114 3 6 33 25 33 636 312 960 870 871 872 3 6 33 26 69 948 300 624 763 762 764 3 7 34 0 875 680 356 32 289 290 288 3 7 34 1 873 680 356 32 295 296 294 3 7 34 2 680 97 745 421 874 875 873 3 7 34 3 225 896 248 572 295 296 294 3 7 34 4 227 896 248 572 289 290 288 3 7 34 5 549 464 140 788 295 296 294 3 7 34 6 551 464 140 788 289 290 288 3 7 34 7 295 873 225 549 86 84 85 3 7 34 8 296 873 225 549 83 81 82 3 7 34 9 294 873 225 549 88 87 89 3 7 34 10 289 875 227 551 102 104 103 3 7 34 11 290 875 227 551 99 101 100 3 7 34 12 288 875 227 551 107 106 105 3 7 34 13 356 853 529 205 874 875 873 3 7 34 14 32 637 961 313 874 875 873 3 7 34 15 248 565 889 241 226 225 227 3 7 34 16 247 565 889 241 229 228 230 3 7 34 17 572 781 457 133 226 225 227 3 7 34 18 140 925 601 277 550 549 551 3 7 34 19 138 925 601 277 556 557 555 3 7 34 20 788 709 61 385 550 549 551 3 7 34 21 889 675 27 351 246 248 247 3 7 34 22 241 891 567 243 246 248 247 3 7 34 23 601 123 771 447 139 140 138 3 7 34 24 277 879 231 555 139 140 138 3 7 34 25 133 603 279 927 570 571 572 3 7 34 26 385 195 519 843 787 786 788 3 8 35 0 802 704 380 56 508 509 507 3 8 35 1 803 704 380 56 505 506 504 3 8 35 2 704 413 89 737 801 802 803 3 8 35 3 155 920 272 596 505 506 504 3 8 35 4 154 920 272 596 508 509 507 3 8 35 5 479 164 812 488 505 506 504 3 8 35 6 478 164 812 488 508 509 507 3 8 35 7 505 803 155 479 426 427 428 3 8 35 8 506 803 155 479 423 424 425 3 8 35 9 504 803 155 479 431 430 429 3 8 35 10 508 802 154 478 418 417 419 3 8 35 11 509 802 154 478 415 414 416 3 8 35 12 507 802 154 478 420 422 421 3 8 35 13 380 197 845 521 801 802 803 3 8 35 14 56 953 305 629 801 802 803 3 8 35 15 272 881 233 557 153 155 154 3 8 35 16 271 881 233 557 156 158 157 3 8 35 17 596 125 773 449 153 155 154 3 8 35 18 812 53 701 377 477 479 478 3 8 35 19 810 53 701 377 483 484 485 3 8 35 20 488 809 161 485 477 479 478 3 8 35 21 233 894 246 570 270 272 271 3 8 35 22 557 462 138 786 270 272 271 3 8 35 23 701 90 738 414 811 812 810 3 8 35 24 377 846 198 522 811 812 810 3 8 35 25 449 822 498 174 594 595 596 3 8 35 26 485 162 486 810 487 486 488 

1. A multi-node computer system comprising: a plurality of computer nodes interconnected via a Kautz topology having order O, diameter n, and degree k; wherein the order O=(k+1)k^(n−1); wherein data interconnections from a computer node x to a computer node y in the Kautz topology satisfies a relationship y=(−x*k−j) mod O, where 1≦j≦k; wherein each x,y pair includes a unidirectional control link from computer node y to computer node x to convey flow control and error information from a receiving computer node y to a transmitting computer node x; and wherein j represents a corresponding node in the multi-node computer system.
 2. The multi-node computer system of claim 1 wherein a receiving computer node y transmits control packets on a control link to transmitting computer node x to identify a last correctly received data packet from the transmitting computer node, and to identify whether an error has been detected during transmission of the received data packet.
 3. The multi-node computer system of claim 2 wherein a transmitting computer node x stores transmitted packets and keeps the transmitted packets available for replay in response to control messages on the control link.
 4. The multi-node computer system of claim 1 wherein a receiving computer node y transmits buffer status information to a transmitting computer node x to identify buffer availability of downstream computer nodes.
 5. The multi-node computer system of claim 4 wherein a transmitting computer node x includes logic to transmit a packet downstream only if all necessary buffers are available.
 6. A multi-node computer system comprising: a plurality of computer nodes interconnected via a Kautz topology having order O, diameter n, and degree k; wherein the order O=(k+1)k^(n−1); wherein data interconnections from a computer node x to a computer node y in the Kautz topology satisfies a relationship y=(−x*k−j) mod O, where 1≦j≦k; wherein each x,y pair includes a unidirectional control link from computer node y to computer node x to convey flow control and error information from a receiving computer node y to a transmitting computer node x; and wherein j is an integer value.
 7. A multi-node computer system comprising: a plurality of computer nodes interconnected via a Kautz topology having order O, diameter n, and degree k; wherein the order O=(k+1)k^(n−1); wherein data interconnections from a computer node x to a computer node y in the Kautz topology satisfies a relationship y=(−x*k−j) mod O, where 1≦j≦k; wherein each x,y pair includes a unidirectional control link from computer node y to computer node x to convey flow control and error information from a receiving computer node v to a transmitting computer node x; and wherein the plurality of computer nodes in the Kautz topology are numbered from zero to O-1.
 8. A multi-node computer system comprising: a plurality of computer nodes interconnected via a Kautz topology having order O, diameter n, and degree K wherein the order O=(k+1)k^(n−1); wherein data interconnections from a computer node x to a computer node y in the Kautz topology satisfies a relationship y=(−x*k−j) mod O, where 1≦j≦k; wherein each x,y pair includes a unidirectional control link from computer node y to computer node x to convey flow control and error information from a receiving computer node y to a transmitting computer node x; wherein the Kautz topology is uniformly tiled based on one-to-one mapping in accordance with equation: V _(G) →I×V _(T) wherein V_(G) is a set of vertices associated with the Kautz topology; wherein V_(T) represents a set of vertices of a tile; wherein I is an index set; and wherein the tile includes multiple computer nodes.
 9. The multi-node computer system of 8, wherein a given x,y pair is an edge within a tile and t⁻¹ (i,x), t⁻¹ (i,y) is an edge of the Kautz topology.
 10. A multi-node computer system comprising: a plurality of computer nodes interconnected via a Kautz topology having order O, diameter n, and degree k: wherein the order O=(k+1)k^(n−1); wherein data interconnections from a computer node x to a computer node y in the Kautz topology satisfies a relationship y=(−x*k−j) mod O, where 1≦j≦k; wherein each x,y pair includes a unidirectional control link from computer node y to computer node x to convey flow control and error information from a receiving computer node y to a transmitting computer node x; and wherein the Kautz topology is defined by the relationship y=(−x*k−j) mod O, where 1≦j≦k; wherein j is an integer.
 11. A method for defining a multi-node computer environment, the method comprising: creating a computer system to include a plurality of computer nodes interconnected via a Kautz topology having order O, diameter n, and degree k; wherein the order O=(k+1)k^(n−1); wherein data interconnections from a computer node x to a computer node y in the Kautz topology to satisfy relationship y=(−x*k−j) mod O, where 1≦j≦k; and providing each x,y pair to include a unidirectional control link from computer node y to computer node x, the unidirectional control I ink conveying flow control and error information from a receiving computer node y to a transmitting computer node x; and wherein j is an integer value.
 12. The method of claim 11, wherein j represents a corresponding node in the multi-node computer system.
 13. The method of claim 11, wherein the plurality of computer nodes in the Kautz topology are numbered from zero to O-1.
 14. The method of claim 11, wherein the Kautz topology is uniformly tiled based on one-to-one mapping in accordance with equation: V _(G) →I×V _(T) wherein V_(G) is a set of vertices associated with the Kautz topology; wherein V_(T) represents a set of vertices of a tile; wherein I is an index set; and wherein the tile includes multiple computer nodes.
 15. The method of claim 14, wherein a given x,y pair is an edge within a tile and t⁻¹ (i,x), t⁻¹ (i,y) is an edge of the Kautz topology.
 16. The method of claim 11, wherein the Kautz topology is defined by the relationship y=(−x*k−j) mod O, where 1≦j≦k; wherein j is an integer. 